Introduction | Expressing | Measurement | Calculations | Graphs | Non-linear relationships | Investigations

## Expressing uncertainty

There are two ways that the uncertainty in a quantity can be expressed – as an absolute value and as a percentage.

## Absolute uncertainty

Absolute uncertainty refers to the raw margin of error in any measurement (i.e. the "give or take").

__Example:__A car is measured to be 2.8 m long, give or take 0.1m. We can write the length of the car as:

L = 2.8 m ± 0.1 m

Where 0.1m is said to be the absolute uncertainty.

__Example:__John is between 174.5 cm and 175.5 cm tall. Another way of thinking of this, is that John's height is within 0.5cm of 175cm. We can therefore write John's height as:

Height = 175 cm ± 0.5 cm

Where 0.5cm is the absolute uncertainty.

## Percentage uncertainty

It is often useful to write the uncertainty as a percentage of the measurement. If we measure something to be

*, with an absolute uncertainty of***x****Δ**, then:*x***Percentage uncertainty = (**

*Δx*/*x*) × 100%

__Example:__In the example above the car was measure to be 2.8m long with an absolute uncertainty of 0.1m. Since 0.1m is 4% of 2.8 m, the percentage uncertainty in the cars length is 4%. We therefore can write the length of the car with the percentage uncertainty as:

L = 2.8 m ± 4%

## Rounding uncertainties

**Round both absolute and p****ercentage**uncertainties to 1 s.f.**Then round the measurement to the same number of decimal places as the**__absolute uncertainty__

__Example:__2.5764 ± 0.8453 (unrounded)

*Written as absolute uncertainty:*

2.6 ± 0.8

*Written as percentage uncertainty:*

(0.8453/2.5764) × 100% = 32.81% = 30% (1sf)

2.6 ± 30%

**Example:**1234 ± 456 (unrounded)

*Written as absolute uncertainty:*

1200 ± 500

*Written as percentage uncertainty:*

(456/1234) × 100% = 36.95% = 40% (1sf)

1200 ± 40%